Almost coinciding families and gaps in \({\mathcal P}(\omega)\)
The author studies nontrivial almost coinciding families indexed by \(\omega^ \omega\), abbreviated by \(F\). A. Dow et al. have shown that the existence of such an \(F\) is independent of the negation of the continuum hypothesis, CH. MA is Martin's axiom. It is shown that the following additional axioms imply the non-existence of a family \(F\): (a) If \(P\) is a poset which adjoins \(\omega_ 2\) Cohen reals, then in \(V^ P\) such an \(F\) does not exist. (b) Assume \(\omega_ 1< \kappa= \kappa^{<\kappa}\) (forcing notation). There exists a poset \(P\) with the countable chain condition such that in \(V^ P\) holds \(2^ \omega= \kappa\) + MA + there exists an unfilled \((\kappa, \kappa)\)- gap + such an \(F\) does not exist. The author poses the question whether or not ``ZFC + \(d>\omega_ 1\) + there exists such an \(F\)'' is consistent.
poset with the countable chain condition
scientific article; zbMATH DE number 469067